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Essentials of Mathematical Statistics

Book Description

Part of the Jones and Bartlett Learning International Series in Mathematics Written for the one-term introductory probability and statistics course for mid- to upper-level math and science majors, Essentials of Mathematical Statistics combines the topics generally found in main-stream elementary statistics books with the essentials of the underlying theory. The book begins with an axiomatic treatment of probability followed by chapters on discrete and continuous random variables and their associated distributions. It then introduces basic statistical concepts including summarizing data and interval parameter estimation, stressing the connection between probability and statistics. Final chapters introduce hypothesis testing, regression, and non-parametric techniques. All chapters provide a balance between conceptual understanding and theoretical understanding of the topics at hand. Key Features of Essentials of Mathematical Statistics: - End-of-section exercises range from computational to conceptual to theoretical. - Many sections include a sub-section titled “Software Calculations” which gives detailed descriptions of how to perform the calculations discussed in the section using the software Minitab, R, Excel, and the TI-83/84 calculators. - Provides a clear balance between conceptual understanding and theoretical understanding - Exercises throughout vary in level of difficulty and scope.

Table of Contents

  1. Cover
  2. Title Page
  3. Copyright
  4. Contents
  5. Preface
  6. 1 Basics of Probability
    1. 1.1 Introduction
    2. 1.2 Basic Concepts and Definitions
    3. 1.3 Counting Problems
    4. 1.4 Axioms of Probability and the Addition Rule
    5. 1.5 Conditional Probability and the Multiplication Rule
    6. 1.6 Bayes’ Theorem
    7. 1.7 Independent Events
  7. 2 Discrete Random Variables
    1. 2.1 Introduction
    2. 2.2 Probability Mass Functions
    3. 2.3 The Hypergeometric and Binomial Distributions
    4. 2.4 The Poisson Distribution
    5. 2.5 Mean and Variance
    6. 2.6 Functions of a Random Variable
    7. 2.7 The Moment-Generating Function
  8. 3 Continuous Random Variables
    1. 3.1 Introduction
    2. 3.2 Definitions
    3. 3.3 The Uniform and Exponential Distributions
    4. 3.4 The Normal Distribution
    5. 3.5 Functions of Continuous Random Variables
    6. 3.6 Joint Distributions
    7. 3.7 Functions of Independent Random Variables
    8. 3.8 The Central Limit Theorem
    9. 3.9 The Gamma and Related Distributions
    10. 3.10 Approximating the Binomial Distribution
  9. 4 Statistics
    1. 4.1 What Is Statistics?
    2. 4.2 Summarizing Data
    3. 4.3 Maximum Likelihood Estimates
    4. 4.4 Sampling Distributions
    5. 4.5 Confidence Intervals for a Proportion
    6. 4.6 Confidence Intervals for a Mean
    7. 4.7 Confidence Intervals for a Variance
    8. 4.8 Confidence Intervals for Differences
    9. 4.9 Sample Size
    10. 4.10 Assessing Normality
  10. 5 Hypothesis Testing
    1. 5.1 Introduction
    2. 5.2 Testing Claims About a Proportion
    3. 5.3 Testing Claims About a Mean
    4. 5.4 Comparing Two Proportions
    5. 5.5 Comparing Two Variances
    6. 5.6 Comparing Two Means
    7. 5.7 Goodness-of-Fit Tests
    8. 5.8 Test of Independence
    9. 5.9 One-Way ANOVA
    10. 5.10 Two-Way ANOVA
  11. 6 Simple Regression
    1. 6.1 Introduction
    2. 6.2 Covariance and Correlation
    3. 6.3 Method of Least Squares
    4. 6.4 The Simple Linear Model
    5. 6.5 Sums of Squares and ANOVA
    6. 6.6 Nonlinear Regression
    7. 6.7 Multiple Regression
  12. 7 Nonparametric Statistics
    1. 7.1 Introduction
    2. 7.2 The Sign Test
    3. 7.3 The Wilcoxon Signed-Rank Test
    4. 7.4 The Wilcoxon Rank-Sum Test
    5. 7.5 The Runs Test for Randomness
  13. A Proofs of Selected Theorems
    1. A.1 A Proof of Theorem 3.7.5
    2. A.2 A Proof of the Central Limit Theorem
    3. A.3 A Proof of the Limit Theorem of De Moivre and Laplace
    4. A.4 A Proof of Theorem 4.6.1
    5. A.5 Confidence Intervals for the Difference of Two Means
    6. A.6 Coefficients in the Linear Regression Equation
    7. A.7 Wilcoxon Signed-Rank Test Distribution
  14. B Software Basics
    1. B.1 Minitab
    2. B.2 R
    3. B.3 Excel
    4. B.4 TI-83/84 Calculators
  15. C Tables
  16. D Answers to Selected Exercises
  17. Index